Circle In An Isosceles Triangle

This type of problem has been showing up more and more often on the Calculator test. I have personally seen it on a region test, district test, and a few invitational tests.

If the circle is inside the isosceles triangle, then the radius can be found using 1/2*b*sqrt((2a-b)/(2a+b)) where a is the smallest side and b is one of the congruent sides.

-Kevin

Law of Cosines

This is typically used for problem number 40.

When given a triangle with two sides a and b and the included angle, the remaining side can be calculated using the law of cosines:   

Example:     

 

-Kevin

Scaling Principles

 Principle 1: LengthIf x and y are length dimensions on geometrically similar figures, then y is proportional to x, then
 

Principle 2: Area
If x is a length and A is an area on geometrically similar figures, then

Principle 3: Volume

If x is a length and V is a volume on geometrically similar figures then

Example:

xin/9.4in=273cm/100cm = 25.662 in shadow * 2.54*10 = 652 mm (2.54 cm in 1 inch)  

Kevin Valle :D

Percents

Percent Change/Difference = 100 * [(B/A) - 1] where B is the 2nd number given and A is the 1st.

Example:

 

B = area of letter-sized sheet = 8.5*11 = 93.5
A = area of legal-sized sheet = 8.5*13 = 110.5

% change = 100 * [(93.5/110.5)-1] = -15.4 or -1.54x10^1

Percent Error = 100 * [(approximate/exact) -1]

Example:

Approximate = 1152
Exact = 1176.37
Percent Error = 100 * [(1152/1176.37)-1] = -2.07 (notice that this is an SD problem and is 3 sig figs)

-Kevin